You can prevent a student with a balance due from registering or making a purchase on ASAP.
From the left nav panel, select 1 Preferences and in the Payment and Transaction Configuration section, set 2 Force payment of balance due to Yes.
Once a student is logged in, if a student with a balance due selects a class (or an event/conference/makes a reservation/buys a product), the student will be brought immediately to a 3 statement of account page and will be unable to add any new items to the shopcart before paying for the unpaid invoices. The student will not be able to make any new purchases before paying the existing balance.
On Admin, you will see a 4 message in QuickReg that alerts you to the fact that the student has a balance due. On admin, you have the option to continue with the registration with the balance due and to navigate to the student details screen, select the unpaid invoice(s) and make payments.